This is known as "scalar gauge invariance" [1]}, [2]}, [3]} because the gauge transformation parameter is an arbitrary scalar \(\phi (x)\) . In fact, (REF ) is the equation of motion of a conformal symmetric tensor \(h_{ab}\)  [3]} with gauge invariance given by (REF ). So in some sense \(h_{ab}(x)\)  can be thought of as a conformal symmetric tensor living on the celestial sphere. Again, the "gauge-transformation" has no obvious origin except that this is somehow related to supertranslation at null-infinity which is also determined by a single function [5]}, [6]}. Clarification of this connection seems to be of utmost importance. Let us now get rid of the scalar contribution to \(O(x)\)  using this "redundancy".
